Effect of atomic scale plasticity on hydrogen diffusion in iron: Quantum mechanically informed and on-the-fly kinetic Monte Carlo simulations
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Effect of atomic scale plasticity on hydrogen diffusion in
iron: Quantum mechanically informed and on-the-fly kinetic
Monte Carlo simulations
A. Ramasubramaniam
Program in Applied and Computational Mathematics, Princeton University,
Princeton, New Jersey 08544
M. Itakura
Center for Computational Science and E-systems, Japan Atomic Energy Agency,
Taito-ku, Tokyo 110-0015, Japan
M. Ortiz
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125
E.A. Cartera)
Department of Mechanical and Aerospace Engineering and Program in Applied and Computational
Mathematics, Princeton University, Princeton, New Jersey 08544
(Received 25 February 2008; accepted 2 July 2008)
We present an off-lattice, on-the-fly kinetic Monte Carlo (KMC) model for simulating
stress-assisted diffusion and trapping of hydrogen by crystalline defects in iron. Given
an embedded atom (EAM) potential as input, energy barriers for diffusion are
ascertained on the fly from the local environments of H atoms. To reduce
computational cost, on-the-fly calculations are supplemented with precomputed
strain-dependent energy barriers in defect-free parts of the crystal. These precomputed
barriers, obtained with high-accuracy density functional theory calculations, are used to
ascertain the veracity of the EAM barriers and correct them when necessary. Examples
of bulk diffusion in crystals containing a screw dipole and vacancies are presented.
Effective diffusivities obtained from KMC simulations are found to be in good
agreement with theory. Our model provides an avenue for simulating the interaction of
hydrogen with cracks, dislocations, grain boundaries, and other lattice defects, over
extended time scales, albeit at atomistic length scales.
I. INTRODUCTION embrittlement (HE) of metals, hydrogen-enhanced deco-
The interaction of hydrogen with metals is a long- hesion (HEDE)4–6 and hydrogen-enhanced local plastic-
standing problem of interest across several disciplines ity (HELP)2,7,8 have gained acceptance as the two most
such as surface chemistry and catalysis, applied physics, viable for stable phases. Material degradation via the
metallurgy, and mechanical engineering. The deleterious formation of brittle hydride phases,9 possibly stabilized
effect of hydrogen on the mechanical properties of metals by local stresses,10 is also a viable mechanism but has not
and alloys remains a vexing problem for structural ap- been found to be relevant for iron,1 which is the material
plications and is still incompletely understood from a of interest in this work. The HEDE mechanism postulates
mechanistic standpoint due to the multiplicity of mecha- embrittlement due to localized reduction in cohesive
nisms by which hydrogen interacts with metals. Further- strength induced by the segregation of hydrogen to de-
more, it is now widely accepted that several of these fects such as grain boundaries, microcracks, notches, and
phenomena are inherently coupled1–3 and the search for second phase particles, among others. In this case, em-
a single dominant mechanism, which has fueled much brittlement is directly attributable to a decrease in the
research and (inevitably) opposing viewpoints, is futile. strength of atomic bonding of the host metal in regions of
Among several mechanisms proposed for hydrogen hydrogen accumulation. The HELP mechanism is based
on the observation that hydrogen, which tends to form a
Cottrell atmosphere11 around dislocation cores, reduces
a)
Address all correspondence to this author. barriers to dislocation motion and effectively unpins dis-
e-mail: eac@princeton.edu locations at lower stress levels. This leads to highly
DOI: 10.1557/JMR.2008.0340 localized slip in the vicinity of crack surfaces and
J. Mater. Res., Vol. 23, No. 10, Oct 2008 © 2008 Materials Research Society 2757A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations eventually to localized plastic failure. In this sense, features that seem most pertinent to HEDE and HELP HELP is more of a strength degradation mechanism mechanisms, with emphasis upon the microscopic proc- rather than embrittlement, which is commonly perceived esses involved. For HEDE, the relevant microscopic to be a toughness degradation mechanism. process is the reduction in cohesive strength of the ma- In broad terms, HE models require a description of (i) terial in regions of hydrogen accumulation. Recent first- hydrogen transport within metals, inevitably in the pres- principles calculations by van der Ven and Ceder30 for Al ence of microstructural imperfections, and (ii) the effect (111) surfaces and by Jiang and Carter31 for Al (111) and of hydrogen on the mechanical behavior of the host bcc Fe (110) surfaces clearly demonstrate a reduction metal, again mediated by interactions with defects. These both in surface energy as well as cohesive strength of issues have been extensively discussed in compilations these interfaces with increasing hydrogen coverage. This and reviews1,12–17 and we only briefly outline the basic implies that segregation of hydrogen to crack tips (within mechanisms with emphasis on ␣–iron [body-centered cu- the so-called “cohesive zone” region ahead of a crack), or bic (bcc)] where necessary. The introduction of hydrogen to other internal interfaces within the material, will lead from the environment can occur by proton deposition at to cleavage along fracture planes or decohesion of inter- a free surface or by dissociative adsorption. An adsorbed faces at lower levels of load. Indeed, this model for re- hydrogen atom can be incorporated subsequently in a duced cohesive strength with increasing hydrogen cov- subsurface interstitial site and diffuse thereafter into the erage, with proper upscaling to continuum scales via bulk. In the case of bcc Fe (110) and Fe (100) surfaces, renormalization,32,33 was used in the work of Serebrin- density functional theory (DFT) calculations18,19 clearly sky et al.34 and shown to produce plausible results for indicate that the adsorption process is exothermic (i.e., it intermittent crack propagation in embrittled steels. Simi- is energetically preferable for hydrogen to adsorb on the lar models, albeit with a phenomenological law for deg- Fe surface than remain in a molecular state) while incor- radation of cohesive strength, have also been pro- poration into subsurface sites is endothermic (i.e., hydro- posed.35,36 The interaction of hydrogen with defects in gen would rather remain on/segregate to a free surface crystalline materials is reviewed at length in Ref. 14. than be in the bulk). Once incorporated in the bulk, hy- With reference to the HELP mechanism, interaction of drogen diffuses rapidly between tetrahedral sites of the hydrogen with dislocations is of prime importance. Di- bcc lattice, the diffusivity being on the order of 10−8 m2/s rect observations of enhanced dislocation mobility in- in a perfect lattice.19,20 However, microstructural imper- duced by hydrogen,8 even under conditions of constant fections (vacancies, solute atoms, dislocations, grain stress, suggest that hydrogen shields the elastic fields of boundaries, etc.) provide low energy trapping sites dislocations and allows for extensive localization of slip, within the lattice and retard the overall diffusion.13,21–26 which is typically observed along fracture surfaces in The existence of these traps plays an important role in the hydrogen embrittled iron.2 HELP has been modeled at a retention of hydrogen within the bulk, given the low continuum level by prescribing a local flow stress that solubility of hydrogen in iron.1 Furthermore, when trap- decreases with increasing hydrogen concentration37; the ping occurs at interfaces such as matrix/second-phase constitutive law for softening is not obtained from the particles, grain boundaries, microcracks, etc., the cohe- actual material response and is a phenomenological con- sive strength of these interfaces is compromised, thereby struct designed to reproduce macroscopic experimental promoting decohesion and/or emission of dislocations. observations. Molecular dynamics studies of the interac- Last, but not least, in describing bulk diffusion of hydro- tion of a single hydrogen atom38 with an edge dislocation gen we must account for its interaction with stress fields in bcc iron suggest that hydrogen dissolved in tetrahedral that arise from existing lattice defects or from external sites can reduce the Peierls stress for dislocation motion. loads.11,27,28 It is generally assumed that hydrogen seg- Conversely, the same studies also suggest that disloca- regates preferentially in dilatational strain fields:29 later tions can be pinned upon encountering hydrogen– in this article, we provide results from DFT calculations vacancy complexes. Recent atomistic studies39 on the that clearly quantify this effect. The concentration energetics of kink-pair nucleation at screw dislocations gradient, which provides the driving force for diffusion, suggest that hydrogen can decrease the kink-pair forma- is sensitive to stress gradients. Thus, hydrogen in solu- tion energy upon jumping to the strong binding site at a tion will be driven preferentially towards regions of di- dislocation core. Hydrogen that is already bound at a latational strain (such as crack tips) thereby inducing dislocation core can, however, impede the motion of ex- hydrogen-enhanced decohesion or plasticity at these lo- isting kinks. At higher temperatures, hydrogen jumps cations. more frequently to dislocation cores, inducing kink-pair The effect of hydrogen on the mechanical properties of nucleation, and also jumps away from dislocation cores metals has been well documented in the reviews noted frequently enough to not impede kink motion. These above; in particular, details relevant to iron and steel may findings would appear to be consistent with experiments be found in the review by Hirth.1 We note a few salient that report hydrogen-induced softening in high purity 2758 J. Mater. Res., Vol. 23, No. 10, Oct 2008
A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
iron, believed to be mediated by screw dislocation mo- nomenological assumptions inherent in higher length
tion, at temperatures higher than 200 K.40,41 scale models.
From the above discussion, which is certainly a re- With the preceding remarks in mind, we outline the
stricted view of embrittlement mechanisms, it is already major goals of this work. It is fairly self-evident that
evident that hydrogen–metal interactions are remarkably hydrogen–metal interactions are complex and a compre-
complex to model. There has been extensive research on hensive understanding of embrittlement will entail a
modeling these interactions from quantum mechanical to more detailed reckoning than can be provided by phe-
continuum levels. Some of these studies have already nomenology. Some progress can be made by incorporat-
been alluded to before and we note next a few more ing information from more accurate calculations (quan-
works that have sought to model the overall embrittle- tum mechanical/atomistic) at smaller length scales in
ment process rather than individual mechanisms. At the continuum models. As noted previously, recent work by
continuum level, Sofronis and coworkers37,42–44 have Carter, Ortiz, and coworkers31,33,34 along these lines has
used finite element studies extensively to model the in- provided a basis for addressing crack propagation by
teraction of hydrogen with cracks and notches. A brief hydrogen-enhanced decohesion. Although that work
survey with further references may be found in Ref. 44. used a cohesive law from quantum mechanical calcula-
The primary features of these models are stress-assisted tions, crack tip plasticity and stress-assisted diffusion
diffusion and an elasto-plastic constitutive law (J2 were still treated at a phenomenological level. It is not
plasticity) for material response; additional constitutive immediately obvious how this situation may be remedied
assumptions need to be made to capture hydrogen- within a continuum approach. On the other hand, a fully
induced dilatation, trap generation with plastic strain, and discrete approach, while computationally more expen-
changes in local elastic moduli with hydrogen con- sive, provides an immediate avenue for progress. In this
centration. The main outcomes of such studies are pre- work, we focus on this particular issue of developing
dictions related to initiation of fracture from or near discrete approaches to model hydrogen–metal interac-
notches, necking instabilities and shear localization, and tions in metals, ␣-iron being the candidate metal. The
macroscopic stress–strain behavior, among others. As mechanics of hydrogen interaction with the host metal
with all continuum models, the validity of the constitu- and microstructural defects, which are now fully atom-
tive assumptions can be established only to the extent istically resolved, is relatively easy to describe with an
that the observed behavior is not contrary to direct ex- appropriate interatomic potential. We use an EAM po-
perimental evidence. Other models that have a more mi- tential for iron and hydrogen developed by Wen et al.46
cromechanical basis have also been proposed to analyze in this work. The main difficulty lies in modeling the
the interaction of discrete dislocation pile-ups with temporal aspect of the problem, since processes such as
cracks in the presence of hydrogen (e.g., Refs. 3, 45, and hydrogen absorption and bulk diffusion (with trapping)
references therein). Such models allow for a more direct are thermally activated processes and, hence, are rare
description of embrittlement in the presence of crack tip events that cannot be simulated over reasonable compu-
plasticity and hydrogen. It should be noted that the tational time scales with standard molecular dynamics.
discrete dislocation modeling is at the level of linear As an alternative, we present a kinetic Monte Carlo (KMC)
elasticity, which precludes a description of the disloca- approach, which can achieve extended time scales for
tion core and possible interactions of consequence long-range hydrogen diffusion. Since our interest is in
with hydrogen, and stress-assisted diffusion is still describing both normal lattice diffusion as well as diffu-
treated phenomenologically. More recently, there have sion in strained and defective crystals, a lattice-based
been fully three-dimensional (3D) atomistic studies46 model will not suffice. We have developed an off-lattice
on crack propagation in hydrogen embrittled ␣-iron us- model within which barriers to diffusion are ascertained
ing molecular dynamics (MD) with embedded atom on the fly with the sole knowledge of the local environ-
(EAM) potentials.47–49 Qualitatively, these studies pre- ment of the diffusing atom. Thus, unlike standard event-
dict blunting by dislocation emission of crack tips in list-based KMC models, our model is applicable even
the absence of hydrogen whereas hydrogen charged when hitherto unforeseen configurations are encoun-
specimens show cleavage with some plasticity. The tered. Furthermore, diffusion barriers are now stress-
drawbacks with such MD calculations are the application dependent, since these are computed directly from the
of unphysically high strain rates (e.g., ∼10 10 /s in local atomic configuration. To speed up the on-the-fly
Ref. 46), which entirely preclude experimental corrobo- calculations, which are all but unnecessary, except when
ration of predictions, as well as the lack of long- local strains are large or the lattice defective, we carry
range diffusion of hydrogen in the bulk due to small out supplementary DFT50,51 calculations to precompute
simulation times (∼0.4 ns). Nevertheless the ability to strain-dependent barriers for small to moderate lattice
model material response from an interatomic potential is strains. These DFT calculations, which are of much
still attractive from the perspective of eliminating phe- higher accuracy and reliability than empirical potentials,
J. Mater. Res., Vol. 23, No. 10, Oct 2008 2759A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
are also used to verify the reliability of the EAM poten- the cell vectors should not lead to significant interactions
tial, which is fit only to bulk equilibrium properties.46 between periodic images of H atoms. For our Fe54H su-
The organization of this article is as follows. Section II percell, the relative change in cell volume is 0.73% and
describes the numerical methods used. Section III pro- 0.72% when H is dissolved in the tetrahedral (T) and
vides results of DFT calculations and of KMC simula- octahedral (O) sites respectively. Furthermore, when
tions of bulk diffusion in the presence of traps arising only ionic relaxation is allowed, the dissolution energy
from microstructural defects (dislocations and vacan- (referenced to the gaseous hydrogen molecule and pure
cies). Section IV provides a summary with directions for bcc Fe), defined here as
future work.
1
Ed = E共Fe54H兲 − E共Fe54兲 − E共H2兲 , (1)
II. MODELING APPROACH 2
A. DFT calculations is found to be 0.22 and 0.36 eV for dissolution in the
T-sites and O-sites, respectively; Jiang and Carter19 re-
We perform spin-polarized DFT calculations, within
port values of 0.19 eV (T-site) and 0.32 eV (O-site) when
the generalized gradient approximation (GGA) of the
both ionic and cell-vector relaxation is allowed. Our pres-
Perdew-Burke-Ernzerhof (PBE) form52 for electron ex-
ent result for the dissolution energy of H in T-sites is thus
change and correlation, using the Vienna Ab Initio Simu-
0.03 eV higher than Jiang and Carter’s result; numerical
lation Package (VASP).53–55 We use the Blöchl projector-
errors being ∼0.02 eV, we deem this to be an acceptable
augmented wave (PAW) method,56 an all-electron frozen
result. Within this level of expected error, our basic con-
core DFT technique, as provided in VASP.57 The PAW
clusion (discussed subsequently in Sec. III) remains un-
potentials represent the nuclei plus core electrons up
changed, namely, the O-site is always higher in energy
through the 3p shell. Detailed calibrations for Fe and H
than the T-site. The O-site energies, therefore, do not
using PAW-DFT-GGA have been reported previously in
play any further role in parameterizing our KMC calcu-
Ref. 19. All our calculations are carried out using a su-
lations and, hence, the slight discrepancy in the predicted
percell containing 54 Fe atoms and 1 H atom, denoted
dissolution energies is inconsequential. Thus, the Fe54H
henceforth as Fe54H. This choice corresponds to a H
supercell is adequately large to minimize interactions be-
concentration of 1.8 at.%. We determined from conver-
tween periodic image H atoms and simultaneously pro-
gence studies a kinetic energy cutoff of 500 eV and a 6 ×
duce well-converged dissolution energies.
6 × 6 k-point sampling for this supercell, which converges
To locate minimum energy pathways between T-sites,
the total energy of the Fe54 cell to within 1 meV/atom.
we use the climbing image nudged elastic band (CI-NEB)59
The dissolution energy of H in a tetrahedral site is con-
method. In this scheme, which is a modification of the
verged to within 2 meV (without atomic relaxation). Bril-
NEB method,60 the highest-energy image climbs uphill
louin zone integration is performed using the first-order
to the saddle point. Hence, fewer images can be used in
Methfessel–Paxton method,58 with a Fermi surface
the elastic band without loss of resolution at the saddle
smearing of 0.1 eV. All structural relaxations are per-
point. For the CI-NEB calculations, we use a spring
formed until forces on atoms are below 0.01 eV/ Å.
force-constant of 5 eV/Å2 between images and relax all
Although large supercells are preferable to minimize
ionic positions until forces on atoms are lesser than 0.01
elastic interactions between interstitial H atoms in ad-
eV/Å. The rank of the saddle point is determined by
joining (periodic neighbor) cells, computational consid-
diagonalizing a Hessian matrix with displacements of
erations impose a limit on practically feasible cell sizes.
0.02 Å; only the H atom is allowed to move in the Hes-
One way to circumvent this difficulty and still use small
sian construction. The much more massive Fe atoms
cell sizes is to fully relax both supercell vectors as well
couple only weakly to the H atom and hence their con-
as ionic positions as was done by Jiang and Carter.19
tribution to the Hessian describing H atom motion can be
Since we wish to parameterize dissolution energies and
safely neglected. Zero-point corrections to T-site and
diffusion barriers as function of lattice strain, the super-
saddle point energies are computed by summing up the
cell must be held fixed at different levels of strain, which
zero-point energies of the real-valued normal vibrational
implies that the cell vectors can no longer be relaxed. If
modes of the H atom as EZPE ⳱ ⌺vihi /2, i being the
we choose our (unstrained) supercell such that relaxation
real-valued frequency.
of cell vectors and ionic positions produces well-
converged H dissolution energies accompanied by a neg-
ligible change in the overall volume, we are assured that B. KMC model for diffusion
interactions between periodic images of H atoms are neg- The case of a defect-free, but not necessarily strain-
ligible even when the cell vectors are held fixed (since free, lattice serves as a good starting point to illustrate the
relaxation will not alter their lengths). It is also reason- details of the KMC model and is discussed first; the
able to expect then that the application of small strains to extension to defective lattices is outlined subsequently.
2760 J. Mater. Res., Vol. 23, No. 10, Oct 2008A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
The precise mechanism of H diffusion is complicated and is, to a very good approximation, independent of
due to the fact that H is a sufficiently light impurity for local strain as demonstrated in Sec. III. In the above
quantum effects to be observable and yet heavy enough expression, Ti are the 3N real-valued normal mode fre-
(as compared to electrons) to be localized in the metal quencies at the T-site and Sj are the 3N − 1 real-valued
lattice; Grabert and Schober61 provide a theoretical re- normal mode frequencies at the saddle point. It is worth
view of these issues. At very low temperatures, coherent noting that the energy barrier is sensitive to strains aris-
tunneling is expected to be the dominant mechanism, ing from other lattice defects or constraints (boundary
giving way to phonon-assisted tunneling as the tempera- conditions) and that this strain-sensitivity is implicitly
ture increases, and eventually to classical over-barrier accounted for in the energy minimization procedure.
jumps at still higher temperatures.17 Hirth1 suggests that With all the events and rates at hand, we apply the n-fold
at temperatures above 300 K diffusion could be de- way KMC algorithm65 to simulate H-diffusion.
scribed by small polaron theory,62 whereas at tempera- The extension of the above procedure to a defective
tures above 800 K, H diffuses in a partly delocalized lattice requires some care. In particular, unlike the case of
manner. In this work, which represents our first attempt a perfect lattice, there is no guarantee that there is actu-
at integrating diffusion kinetics with mechanics, we ig- ally an energy minimum in the interior of a tetrahedron
nore these myriad complications and simply describe dif- within a defective region. In fact, using the H–Fe EAM
fusion by over-barrier hops in the temperature range potential,46 we have found instances of this problem even
300–600 K. for a perfect lattice that is significantly distorted, al-
In ␣-iron, both indirect evidence from experiments,1 as though this could be an artifact of the potential itself. We
well as direct corroboration from DFT calculations,19 illustrate the solution to such problems with a specific
favor dissolution of H in T-sites over O-sites. This result example. Consider an instance where the energy mini-
is valid even when the lattice is subjected to small to mum lies (for whatever reason) not at a T-site but at an
moderate strains (Sec. III). Thus, we need only consider O-site, assuming for now that the lattice is otherwise
hops between T-sites for bulk diffusion. To locate the perfect. The O-site is not in the interior of any tetrahe-
T-sites, we first apply a Delaunay triangulation to the dron; it lies instead on a common edge of four tetrahedra
N-atom bcc lattice, which can generally be implemented [Fig. 1(b)]. It is immediately apparent that proceeding
as an O(N log N) algorithm.63 The Fe atoms are the naively as before to find minimum energy positions
nodes of the triangulated region, while the tetrahedra are within the tetrahedral will lead to degeneracies (within
the volume elements that contain the T-sites of interest. numerical error), as inserting an H-atom in the interior of
The exact position of the dissolved H-atom within the any of these tetrahedra and carrying out an ionic relaxa-
tetrahedron may be determined by relaxing the atoms and tion will lead to it migrating to the O-site. Moreover,
minimizing the energy of the system. From this T-site, relaxing the H-atom on faces containing the common
the H-atom can execute a hop to one of four adjacent edge will also lead to its migration to the same O-site; the
tetrahedra by passing through a common face. From barrier to hops (within numerical error) is then zero.
symmetry arguments, or by a more careful NEB calcu- Thus, in addition to misidentifying the relevant hops, the
lation,19 it may be shown that the saddle point for a hop actual numerical procedure will be corrupted by rapid
lies on the common face between adjoining tetrahedra. back and forth jumps of low activation energy. To avoid
The location of the saddle point is determined by con- such problems in general, and also to correctly identify
straining the H-atom to this face (i.e., by placing the unexpected minimum energy sites when relevant (e.g., H
H-atom on the face and disallowing motion normal to the bound to vacancies at displaced O-sites14,17,66), a pair of
face during relaxation) and again carrying out an energy connected T-sites with (near) zero energy barrier be-
minimization. The barrier for the hop is then the energy tween them are assimilated into a single site. The two
difference between the saddle point and the initial mini- tetrahedra now form a single polyhedron with a single
mum. Once the energy barrier Eb is known, the jump rate energy minimum in its interior, the saddle points being
J at temperature T may be evaluated as J ⳱ J0 exp[Eb/ on the faces of this polyhedron. Multiple tetrahedra can
kT], k being Boltzmann’s constant. The jump frequency be sequentially agglomerated in this manner. For the ex-
J0 can be obtained within harmonic transition-state ample of a low energy O-site, the four tetrahedra are
theory64 from the vibrational normal modes at the saddle combined into a single octahedron with a saddle point on
point and T-site as each of its faces. Our KMC model can thus determine
3N strain-dependent energy barriers for the diffusing species
兿
i=1
T
i
on-the-fly with the local environment of the species as
the sole input.
J0 = 3N−1
, (2) In describing our KMC model thus far, we have pre-
兿
j=1
S
j
sented the most general approach to computing energy
barriers. To construct a computationally efficient model,
J. Mater. Res., Vol. 23, No. 10, Oct 2008 2761A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations though, it is useful to incorporate some a priori knowl- weakly repulsive, which precludes cluster formation. edge about the actual system and keep on-the-fly calcu- Moreover, we will not consider any situations, at present, lations to a minimum. First, it is reasonable to expect that where H2 gas formation may occur (e.g., at internal the H-atom diffuses for the most part in a perfect but cracks or voids). Ignoring H–H interactions is not essen- strained bcc lattice, except when it encounters defects. tial in any way and can be easily dispensed with for Therefore, a lot of computational effort can be saved if diffusion in the bulk; accounting for desorption and for- strain-dependent diffusion barriers can be precomputed mation of H2 molecules would simply require the inclu- and used in these defect-free regions. We demonstrate in sion of additional events in the KMC list. Sec. III, using both DFT and EAM calculations, that simple parameterization of dissolution energies at T-sites and saddle points is indeed possible, at least for small to III. RESULTS AND DISCUSSION: moderate strains. It is therefore possible to switch auto- PARAMETERIZATION OF ENERGY BARRIERS matically from precomputed rates to on-the-fly calcula- AND APPLICATIONS OF KMC TO tions as the diffusing H-atom moves toward a defect, the BULK DIFFUSION transition being determined by a threshold that can be The parameterization of energy barriers as a function adjusted by numerical testing. As an example, a cutoff of of the local strain using PAW–DFT–GGA is presented 2% strain—more precisely, a cutoff of √⑀ij ⑀ij ⳱ 0.02— first and compared to results from the EAM. (To main- implies a switch from precomputed to on-the-fly rates at tain consistency with DFT calculations, all EAM calcu- a distance of about 4 Burgers vectors from the core of a lations reported in this section are also performed on a straight screw dislocation. This is adequate to determine periodic Fe54H cell; the simulation package LAMMPS67 barriers with greater precision near the core of the dis- is employed for this purpose.) Thereafter, the KMC location while reducing computational effort away from model is applied to H diffusion in an ␣–Fe lattice con- the core. Second, note that H is the smallest possible taining vacancies, followed by another example of dif- interstitial atom and that its interaction with Fe atoms is fusion in the presence of a screw dislocation dipole. Ef- fairly localized. Specifically, our DFT calculations for fective diffusion coefficients as a function of temperature the Fe54H supercell with H dissolved in a T-site indicate are extracted from the computations and validated using that the first coordination shell of four Fe atoms moves the standard theory of trapping at lattice defects.13,21,22,26 outward by 0.076 Å, while the second coordination shell moves outward by 0.012 Å. Therefore, it is only neces- sary to relax a ball of atoms around the H-atom, rather A. Diffusion barriers from DFT calculations than the entire computational cell, to locate energy To compute diffusion barriers, we must first ascertain minima and compute dissolution energies. In our calcu- the minimum energy sites and possible pathways be- lations, we use a ball of radius RMD + 2RC centered on tween them. The two candidate minima are the T1 site the H-atom, atoms within RMD ⳱ 6.0 Å being free to (see Fig. 1) and the O-site; we analyze their relative relax while atoms in the outer shell, which is twice the stability as a function of lattice strain. In particular, we cutoff radius RC ⳱ 3.5 Å of the EAM potential, being analyze three distinct deformation modes: volumetric ex- held fixed. This choice roughly corresponds to relaxing pansion (hydrostatic loading), uniaxial tension, and 84 atoms surrounded by 1000 fixed atoms, and we have simple shear. With reference to the axes in Fig. 1, the ascertained that this produces the expected dissolution applied deformation gradient for each of these cases may energies and diffusion barriers. Third, we note that to be expressed as Fh ⳱ (1 + ⑀)I, Fu ⳱ I + ⑀e2 丢 e2, and compute barriers it is necessary only to triangulate the Fs ⳱ I + ␥ e1 丢 e2, respectively, where I is the identity domain locally around each H-atom. Hence, rather than tensor of rank three, ⑀ and ␥ are the applied stretch and triangulate the entire domain at the start, we apply an shear, ei is the unit vector along Cartesian coordinate Xi, incremental Delaunay triangulation, which adds tetrahe- and 丢 denotes the tensor product. Among the various dra sequentially as the H-atom diffuses through the lat- possible choices of uniaxial and shear deformation, the tice. Of course, if the H-atoms move freely through the two indicated here were chosen because these induce the lattice, this eventually amounts to having to triangulate most change in distance between the O-site and the near- the entire domain. On the other hand, if diffusing atoms est (axial) Fe atoms (see Fig. 1). Because H–Fe interac- are strongly trapped in the vicinity of defects, this incre- tions in the bulk are repulsive, these deformations will mental approach can lead to further computational sav- have the greatest influence on the dissolution energy of H ings. Finally, for simplicity, the H–H interaction is taken in the O-site. Table I displays the dissolution energy of H into account only by site-blocking, i.e., disallowing two in T-sites and O-sites as a function of the three deforma- or more hydrogen atoms to occupy the same energy mini- tion modes. It is immediately apparent across the entire mum. This approximation may be justified by noting that range of calculations that the O-site is 0.1–0.2 eV higher the interaction between interstitially dissolved H atoms is in energy than the T-site. Remarkably, the dissolution 2762 J. Mater. Res., Vol. 23, No. 10, Oct 2008
A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
FIG. 1. Schematic illustration of a bcc lattice (a) showing 4 T-sites and 1 O-site on the front face and (b) an exploded view of the tetrahedra that
contain these 4 T-sites. The O-site does not belong to the interior of any tetrahedron and lies instead on the common edge between the 4 tetrahedra.
TABLE I. Dissolution energy Ed ⳱ E(Fe54H) − E(Fe54) − E(H2)/2 (in TABLE II. Energy barrier Eb (meV) and diffusion prefactor D0 (10−7
eV) of hydrogen in the T-site and O-site as a function of lattice strain m2/s) for bulk diffusion of H in ␣–Fe computed as a function of lat-
computed with PAW-DFT-GGA. tice strain. Energies in parentheses represent barriers without
zero-point corrections.
Fh ⳱ (1 + ⑀)I Fu ⳱ I + ⑀e2 丢 e2 Fs ⳱ I + ␥e1 丢 e2
Fh ⳱ (1 + ⑀)I Fu ⳱ I + ⑀ e2 丢 e2 Fs ⳱ I + ␥ e1 丢 e2
⑀ (%) T-site O-site ⑀ (%) T-site O-site ␥ (%) T-site O-site
⑀ (%) Eb D0 ⑀ (%) Eb D0 ␥ (%) Eb D0
−2 0.54 0.68 −2 0.32 0.54 0 0.21 0.35
−1 0.38 0.52 −1 0.27 0.45 1 0.21 0.35 −2 44 (95) 1.87 −2 91 (132) 1.66 0 45 (92) 1.78
0 0.22 0.36 0 0.22 0.36 2 0.21 0.35 −1 46 (94) 1.81 −1 67 (111) 1.72 1 46 (93) 1.80
1 0.08 0.22 1 0.17 0.29 3 0.21 0.35 0 45 (92) 1.78 0 45 (92) 1.78 2 46 (93) 1.79
2 −0.05 0.09 2 0.12 0.22 10 0.22 0.36 1 48 (92) 1.73 1 29 (77) 1.81 3 47 (94) 1.79
2 49 (91) 1.68 2 12 (63) 1.84
energy is unaffected by the application of shear strain,
even up to 10%. For hydrostatic and uniaxial strains, the
dissolution energy increases with compressive strains hop between neighboring T-sites, three intermediate con-
and decreases with tensile strains. Given that H–Fe in- figurations between the initial (T1) and final (T2) state
teractions in the bulk are repulsive, this trend is entirely are created by linear interpolation of atomic positions.
as anticipated. Tensile strains increase the distance be- The supercells are then relaxed using the CI-NEB
tween interstitial sites and the Fe atoms (or, alternatively, method, as noted in Sec. II. A, and the energy barrier
enlarge the “volume” of the interstitial site), thereby low- computed as the difference in energy between the highest
ering the dissolution energy. Conversely, compressive energy intermediate configuration and the initial configu-
strains lead to greater H–Fe repulsion, which raises the ration. Since H is a low mass atom, it is important to
dissolution energy. We may therefore conclude from account for quantum corrections arising from zero point
these results that the T-site is the energetically preferred energy in these calculations of energy barriers; Fe, being
dissolution site for hydrogen in a perfect ␣–Fe lattice much heavier than H, is held fixed in this calculation
over a small to moderate range of strains. (infinite mass assumption). As seen from Table II, al-
Having established site preference for H dissolution in though the overall energy barriers are small, as expected
Fe, we compute energy barriers Eb and prefactors D0 for for a small atom like H, inclusion of zero-point correc-
bulk diffusion, which are displayed in Table II as a func- tions leads to a significant decrease in barriers. The pre-
tion of lattice strain. The diffusion constant at tempera- factor D0, computed using harmonic transition state
ture T is given by the classical Arrhenius expression D ⳱ theory64 and a random walk model for interstitial diffu-
D0 exp[Eb/kT]. To compute the diffusion barrier for a sion in a BCC lattice,68 is given by the expression
J. Mater. Res., Vol. 23, No. 10, Oct 2008 2763A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
3N TABLE III. Dissolution energy of H in Fe at the T-site (Ed) and the
na2
兿
i=1
i
T
na2
saddle point (ES) and the energy barrier (Eb) as a function of lattice
strain. The relative change in circumradius of the tetrahedron (⌬rT)
D0 = = J . (3) and the saddle face (⌬rS), which is a measure of the change in volume
6 3N−1 6 0
兿 S
of the T-site and area of saddle point with applied strain, respectively,
j is also tabulated. For purposes of comparison, the corresponding val-
j=1 ues from the EAM potential are listed in parentheses. All DFT energies
have been corrected for zero-point vibrations.
In this expression, n is the number of equivalent jumps Fh ⳱ (1 + ⑀)I
(four for hops between T-sites in a bcc lattice); a ⳱ a0/√8
⑀, ␥ (%) ⌬rT (%) Ed (eV) ⌬rS (%) ES (eV) Eb (meV)
the jump distance, a0 being the lattice constant of ␣–Fe;
iT the 3N real-valued normal mode frequencies at the −2 −2.00 0.67 (0.54) −2.00 0.71 (0.55) 44 (7)
T-site; and jS the 3N − 1 real-valued normal mode fre- −1 −1.00 0.49 (0.42) −1.00 0.54 (0.44) 46 (22)
0 0.00 0.33 (0.31) 0.00 0.37 (0.34) 45 (33)
quencies at the saddle point. (In principle, D0 should also 1 1.00 0.18 (0.22) 1.00 0.22 (0.26) 48 (41)
contain a vibrational entropy contribution exp[Svib/k]. 2 2.00 0.04 (0.14) 2.00 0.09 (0.18) 49 (46)
We find from our DFT calculations that exp[Svib/k] ≈ 1
and hence omit this term.) Fu ⳱ I + ⑀e2 丢 e2
We observe in Table II that the prefactors are rela- −2 −0.78 0.43 (0.36) −1.32 0.53 (0.48) 91 (115)
tively insensitive to strain and all of these lead to jump −1 −0.39 0.38 (0.34) −0.66 0.45 (0.41) 67 (68)
1 0.40 0.27 (0.26) 0.67 0.30 (0.28) 29 (18)
frequencies on the order of 1013/s. In contrast, the strain
2 0.82 0.22 (0.22) 1.35 0.23 (0.23) 12 (9)
dependence of the energy barriers is clearly discernible
and is of greater importance due to the exponential Fs ⳱ I + ␥e1 丢 e2
dependence of the diffusion constant on the barrier. Just 1 0.29 × 10−3
0.31 (0.31) 0.36 × 10−2 0.36 (0.34) 46 (33)
as the strain dependence of the dissolution energy in the 2 1.19 × 10−3 0.31 (0.31) 1.44 × 10−2 0.36 (0.34) 46 (33)
T-site was explained previously in terms of the change in 3 2.69 × 10−3 0.31 (0.31) 3.25 × 10−2 0.36 (0.34) 47 (33)
volume of the T-site and the accompanying effect on
H–Fe interactions, the strain dependence of energy bar-
riers can be understood by considering both the change in
volume of the T-site and the change in area of the face direct measure of distortion, as a function of applied
containing the saddle point (referred to henceforth as the strain. To compute the relative change in circumradius,
“saddle face”). First, note that the application of shear note that the position vector x of an atom in the deformed
has no effect on the energy barrier; in essence, shearing configuration is expressed in terms of the deformation
the cell does not significantly alter either the volume of gradient F and the position vector X in the undeformed
the T-site or the area of the saddle face. The dissolution configuration as x ⳱ FX. Applying the relevant defor-
energy of H in these sites is therefore unaffected, and mation gradient Fh, Fu, or Fs to the position vectors of the
hence the barrier remains unaltered. Second, in case of vertices of a tetrahedron/face, the circumradius in the
hydrostatic loading, both the area of the saddle face and deformed configuration, and hence the relative change
the volume of the T-site are equally altered. Volumetric with respect to the circumradius in the undeformed con-
expansion lowers the dissolution energy both at the T- figuration, are easily computed.
site and saddle point while compression increases both As seen from the values in Table III, the relative change
these dissolution energies. The synchronous change in in saddle face area as compared to tetrahedron volume is
dissolution energies is such that the net barrier remains most pronounced for uniaxial strains, which explains the
the same. Finally, in case of uniaxial loading, the change stronger strain dependence of the energy barrier. The
in area of the saddle face is more pronounced than the lack of shear strain dependence (up to 10%) of the energy
change in T-site volume. This translates into a stronger barrier is also easy to understand now, given the negli-
effect on saddle point energy than on T-site energy. The gible change in saddle face area and T-site volume during
barrier then shows a clear dependence on the applied shear. For comparison, we also include in Table III en-
strain. ergies computed using the Fe–H EAM potential for a
The qualitative interpretation of strain dependence of periodic Fe54H cell. Since the empirical EAM potential
energy barriers given above can be made more precise by that we use for all on-the-fly KMC calculations is fit only
measuring the actual changes in saddle face area and to bulk properties, a comparison with DFT calculations
T-site volume as functions of applied strain and relating over a range of strains, albeit for defect-free lattices,
these to the dissolution energy at the saddle point and gives us some estimate of the reliability, or lack thereof,
T-site, respectively. Table III displays results from these of the empirical potential.
calculations; note that we report the relative change in The DFT and EAM dissolution data from Table III are
circumradius of the saddle face/tetrahedron, which is a displayed in Fig. 2, as a function of relative change in
2764 J. Mater. Res., Vol. 23, No. 10, Oct 2008A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
FIG. 3. Schematic illustration of tetrahedral site (T1). The vertices, in
FIG. 2. Dissolution energy of H in Fe at the T-site (Ed) and the saddle terms of the lattice constant a0, are V1 ⳱ a0 (1⁄2, −1⁄2, 1⁄2), V2 ⳱ a0 (1⁄2,
1⁄2, 1⁄2), V ⳱ a (0, 0, 1), and V ⳱ (0, 0, 0). Due to tetragonal
point (ES) as a function of the relative change in circumradius of the 3 0 4
tetrahedron (⌬rT) and the saddle face (⌬rS), respectively. The linear symmetry, the dependence of the T-site dissolution energy on strain
fits, which serve as a guide to the eye, indicate that dissolution ener- components ⑀22 and ⑀33 is identical and different from that for ⑀11. The
gies computed from EAM are less sensitive to strain than those from symmetry axis of the face V1V2V3 is denoted by A. The dissolution
DFT calculations. energy at the saddle point on face V1V2V3 is parameterized as a
function of ⑀22 and the strain ⑀AA along axis A (see text).
circumradius. On the whole, it is clear from the data ever, since the on-the-fly calculations rely on the EAM
that the EAM dissolution energies show lesser strain- potential for energy minimization, there will be a discon-
sensitivity than their DFT counterparts. Looking more tinuity when the transition is made from precomputed
closely at the data, we also note that the EAM energy energy barriers to on-the-fly barriers if the former are
barriers are typically within 10–20 meV of the DFT bar- obtained from DFT. Again, referring to Table III and the
riers (although the case of −2% hydrostatic strain discussion in the preceding paragraph, this is not ex-
presents an inexplicably large deviation). Given that (i) pected to result in egregious errors. Nevertheless, for the
DFT energy differences themselves are no more accurate sake of consistency, we use EAM data to carry out the
than ±20 meV and (ii) an error of ±20 meV causes jump parameterization of dissolution energies, thereby mini-
rates to differ by a factor of 1.5–2 over a range of 300– mizing the possibility of discontinuities when making the
600 K, we conclude that the EAM description will likely transition from precomputed to on-the-y barriers. When
suffice for bulk diffusion calculations. Of course, there is fitting the dissolution energy to the diagonal strain com-
no a priori guarantee that the potential is reliable at de- ponents, we have found that a linear fit is too simplistic,
fects; careful case-by-case studies of H-defect energetics and we need to retain at least second-order terms to ob-
are needed for this purpose and should be compared tain energies within 20 meV. For dissolution in the T1
against more accurate methods (such as DFT). We con- site, the dissolution energy is approximated by Ed ⳱
sider a subset of such H–defect interactions further be- 0.3116 − 2.088⑀11 − 4.340(⑀22 + ⑀33) − 60.0(⑀222 + ⑀332),
low. with the strain components referring to the axes indicated
As noted previously, parameterizing the energy barrier in Fig. 3; dissolution energies in the other T-sites follow
(or equivalently the dissolution energies) as a function of from symmetry. Similarly, the saddle point energy on the
strain can lead to significant computational savings. triangular face V1V2V3 shown in Fig. 3 is approximated
From the above results, it is clear that the dissolution by ES ⳱ 0.3486 − 5.937⑀22 − 2.562 ⑀AA, where ⑀AA is a
energy, at least for small strains, is only a function of the tensile strain along the symmetry axis A of the face; the
diagonal components of the strain tensor. Note that the other saddle point energies follow from symmetry. These
tetragonal symmetry of the T-site implies that there are approximations are used when √⑀ij ⑀ij 艋 0.02, in which
two distinct responses for the diagonal components. For case errors engendered are within 20 meV. For √⑀ij ⑀ij >
example, for the T1 site (see Fig. 3) the x2 and x3 direc- 0.02, we switch to on-the-fly calculations with the EAM
tions are equivalent, whereas the x1 direction is distinct. potential.
Hence, the dissolution energy has the same functional
dependence on ⑀22 and ⑀33, and is different from that for B. Bulk diffusion of H with traps
⑀11. We have the choice of using dissolution energies Having discussed the details of the computational pro-
from either DFT or EAM in the fitting procedure. How- cedure and ascertained the quality of the EAM potential
J. Mater. Res., Vol. 23, No. 10, Oct 2008 2765A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
for modeling bulk diffusion, we now present applications and average this over the entire simulation of duration
of our model to diffusion in defective lattices. Specifi- t ⳱ 冱i ti to obtain the diffusion constant D ⳱ 冱i Diti /t.
cally, we first consider diffusion of hydrogen in a lattice Also, the initial positions r(0) for trajectory i + 1 are set
with a screw dislocation dipole followed by diffusion to the final positions r(ti) from trajectory i, which effec-
in a lattice with vacancies. These defects function as tively allows for sampling over different initial condi-
trapping sites for hydrogen, and there exists a well- tions.
established analytical framework21,22 to model the effect
of traps on bulk diffusion. Prior computational studies by 1. Trapping by dislocations
Kirchheim,13,24 who used Monte Carlo simulations to The first example we consider is trapping of diffusing
study interstitial diffusion and trapping in lattices, have H atoms by screw dislocations. We choose to study
already established the validity of these analytical mod- screw dislocations since their dominant strain fields are
els in regimes of low trap occupancy and low H concen- antiplane shear. From the discussion in Sec. III. A, it is
tration. It is not our purpose here to critically analyze clear that such fields will have little or no influence on
these trapping models, but merely to establish that our diffusion barriers, except possibly in the immediate vi-
KMC procedure faithfully reproduces the expected be- cinity of the dislocation core where deformations are
havior within regimes of applicability of these analytical large. Thus the dislocation core itself is the main source
models. Our KMC model is of much wider applicability of trapping and not its associated elastic fields.
and does not require assumptions such as low trap occu- Our simulation cell consists of 106 Fe atoms. The cell
pancy and low H concentration, among others. Addition- vectors are along the [1̄11], [11̄1], and [111̄], directions,
ally, the model automatically accounts for correlations, the cell being 100 Burgers vectors in length along each of
induced by the elastic fields of defects, between saddle these directions. Periodic boundary conditions are im-
point and T-site energies.13,24 posed along the cell vectors; this eliminates the possibil-
In the presence of saturable traps, under conditions of ity of segregation of hydrogen to free surfaces and allows
low hydrogen concentration, low trap occupancy, con- us to focus on bulk diffusion in the presence of traps. To
stant trap density and temperature, and constant saddle maintain periodic boundary conditions, we cannot intro-
point energy, the effective diffusivity of hydrogen may duce a single screw dislocation in the cell and must, at
be estimated as22 minimum, introduce a dipole. An infinitely long screw
冉 冊 −1
dislocation dipole with Burgers vector 1⁄2[1̄11] is intro-
NT duced on a (110) slip plane by displacing the atoms ac-
Deff = DL 1 + KT , (4)
NL cording to the linear elastic solution.72 Using the EAM
potential, the simulation cell is then relaxed in
where DL ⳱ D0 exp[−Eb/kT] is the usual bulk diffusivity
LAMMPS67 via the conjugate gradient method until the
in a perfect lattice, NT is the number of traps per unit
relative change in cell energy is less than 10−4. In its fully
volume, NL is the number of interstitial sites per unit
relaxed state, the screw dislocation core becomes non-
volume, and KT ⳱ exp[−⌬E/kT] is the equilibrium con-
planar and develops threefold symmetry about the [1̄11]
stant for the reversible reaction [H]T-site I [H]trap. The
direction as shown in Fig. 4.
quantity ⌬E is the trap binding energy, which is defined
Recent DFT calculations73,74 indicate that the screw
as the difference between the dissolution energy at the
dislocation core in bcc Fe is sixfold, compact, and non-
trap site and that at a bulk interstitial site. In a bcc lattice,
degenerate in contrast with the threefold, dissociated, and
there are six T-sites per atom: thus, for ␣–Fe, which has
degenerate structure obtained with the EAM potential
a lattice constant at room temperature of 2.8665 Å,69
used in this work. While the distribution of binding sites
NL ⳱ 0.85 mol/cm3.
and binding energies at the core is intimately connected
The diffusion constant D can be measured from a
to the core structure, and thereby the interatomic poten-
KMC calculation by employing the Einstein expression
tial, the on-the-fly method developed here is only con-
1 cerned with locating these binding sites and accurately
D = lim 具关r共t兲 − r共0兲兴2典 , (5) capturing the binding/unbinding of H atoms. Hence, the
t→⬁ 6t
discrepancy between the EAM and DFT core structures
where r is the position vector without wrap-around70 of bears no particular significance in this work.
a diffusing particle and ⟨⭈⟩ denotes the average over all Wen et al.39 have extensively investigated the loca-
particles. Note that the diffusion constant as defined here tion and energies of traps around the threefold, dissoci-
is the tracer diffusion constant, but in the dilute limit, it ated dislocation core; the deepest traps of binding energy
also may be identified with the chemical (Fick’s law) 0.45 eV are indicated schematically in Fig. 4. We find
diffusion constant.17,71 Following Kirchheim,13,24 we these sites to be the main sources of trapping in our KMC
measure the diffusion constant Di ⳱ ⟨[r(ti)–r(0)]2⟩/(6ti) simulation, with other traps being of less significance.
over successive short intervals (or KMC steps) of time ti Hence, there are three traps per length of dislocation line
2766 J. Mater. Res., Vol. 23, No. 10, Oct 2008A. Ramasubramaniam et al.: Atomic scale plasticity on H diffusion in Fe: Quantum mechanically informed and on-the-fly KMC simulations
T-sites into the simulation cell, which corresponds to a
concentration of 7.1 × 10−6 mol/cm3. The maximum pos-
sible fractional occupancies of normal interstitial sites
and trapping sites is NH/(NL − NT) ≈ NH/NL ⳱ 8.3 × 10−6
and NH/NT ⳱ 8.3 × 10−2, respectively. Thus, the con-
centrations and occupancies are dilute enough to use
trapping theory to compute effective diffusivities analyti-
cally. Deviations from trapping theory, if any, must then
be attributable to the existence of multiple traps of dif-
fering energies and possible correlations with saddle
point energies. The diffusion prefactor D0, or equiva-
lently the jump rate J0, is relatively insensitive to strain,
as demonstrated in Sec. III. A. From the data in Table II,
we may therefore infer an average jump rate J0 ⳱ 2.6 ×
1013/s. An experimental measurement of D0 ⳱ 5.12 ×
10−8 m2/s for H diffusion in very pure Fe in the tempera-
ture range 283–348 K,20 yields a jump rate J0 ⳱ 7.6 ×
1012/s. Hence, in this and all subsequent KMC simula-
tions, we set the jump rate to be 1013/s.
Figure 4 displays effective diffusion constants as a
function of temperature both as obtained analytically
from trapping theory [using ⌬E ⳱ −0.45 eV and NT ⳱
8.5 × 10−5 mol/cm3 in Eq. (4)] and from KMC simula-
tions. It is immediately apparent that the analytical and
simulated results are in excellent agreement, and this
provides a first validation of the KMC model. The agree-
ment also suggests that in this instance, the weaker traps
are of less importance, alluded to in the preceding para-
graph, as are correlations between saddle point and in-
terstitial site energies. The simulations were typically run
for 108–109 KMC steps, which correspond to times of
5 × 10−4 to 2 × 10−6 s for temperatures ranging from 300
to 600 K. The diffusivities are usually converged at a
tenth of these times. It is worth noting that such time
scales are well beyond the reach of conventional molecu-
FIG. 4. (a) Differential displacement map of a [1̄11]/(110) screw dis- lar dynamics. Figure 4 also displays the results of a naïve
location core. The dotted lines are a guide to viewing the threefold simulation, which eliminates on-the-fly calculations al-
symmetry. The solid squares indicate, schematically, the three deepest
traps with binding energy of −0.45 eV (after 39). (b) Effective diffu-
together and uses strain-parameterized diffusion barriers
sion constant as a function of temperature in the presence of a screw throughout the lattice. As seen, this approach severely
dipole. The squares represent the measured values from KMC simu- underestimates the diffusion constants (indicating exces-
lations. The solid line represents the calculated values from trapping sively strong binding to the dislocation cores), thereby
theory with ⌬E ⳱ −0.45 eV and 3 trapping sites per Burgers vector. underscoring the utility of the on-the-fly method for ac-
The triangles represent the values obtained from a naïve calculation
curate calculations of energy barriers at defects.
using strain-parameterized diffusion barriers throughout the cell.
Clearly, the latter approach produces grossly incorrect diffusion con- 2. Trapping by vacancies
stants thereby illustrating the need for on-the-fly calculations at de-
fects. The next example we consider is of trapping of H
atoms by vacancies. A vacancy in ␣–Fe is associated
and NT ⳱ 8.5 × 10−5 mol/cm3 ≈ 1025/m3. Kumnick and with six binding sites located along ⟨100⟩ directions be-
Johnson23 estimate trap densities in iron to range be- tween the O-site and the vacancy. Previous experimental
tween 1021/m3 (annealed) to 1023/m3 (60% cold work); and theoretical (effective medium theory) studies have
Oriani22 estimates trap densities in steel to be of the order shown that the binding site is displaced by 0.4 Å from the
of 1025/m3, although it has been suggested26 that this O-site toward the vacancy, with an H-binding energy,
includes the effect of voids induced by hydrogen dam- relative to a bulk T-site, of 0.63 eV.75,76 These studies
age. also indicate that the binding energy is maximum when a
We introduce 50 H atoms (NH ⳱ 50) randomly in single H atom is trapped at a vacancy and decreases with
J. Mater. Res., Vol. 23, No. 10, Oct 2008 2767You can also read
